ANZIAM J.
44 (2003), 485500

An optimal linear filter for random signals with realisations in a separable Hilbert space

P. G. Howlett
Centre for Industrial and Applicable Mathematics
University of South Australia




A. P. Torokhti
Centre for Industrial and Applicable Mathematics
University of South Australia
and
School of Applied Mathematics
The University of Adelaide
Adelaide SA 5005
Australia



Abstract

Let
be a random signal with realisations in an
infinitedimensional vector space
and
an associated observable random signal with
realisations in a finitedimensional subspace
. We seek a pointwisebest estimate of
using a bounded linear filter on the observed
data vector . When
is a finitedimensional Euclidean space and the
covariance matrix for
is nonsingular, it is known that the best
estimate of is given by a standard matrix expression
prescribing a linear meansquare filter. For the
infinitedimensional Hilbert space problem we
show that the matrix expression must be replaced
by an analogous but more general expression using
bounded linear operators. The extension procedure
depends directly on the theory of the Bochner
integral and on the construction of appropriate
HilbertSchmidt operators. An extended example
is given.

Download the article in PDF format (size 117 Kb)


